#163778

order	512
nilpotency class	3
isoclinism	isoclinic class of #163726
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7,8,5)(4,6)(10,14,16,19)(15,21,22,24)(25,26,29,34,30,33,28,32)(27,31,35,38,36,39,37,40);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,19)(13,16)(14,20)(17,22)(18,24)(21,23)(25,27)(26,31)(28,37)(29,35)(30,36)(32,40)(33,39)(34,38);(9,11,13,17)(10,15,16,22)(12,18,20,23)(14,21,19,24)(25,28,30,29)(26,32,33,34)(27,35,36,37)(31,38,39,40);(11,17)(15,22)(18,23)(21,24)(26,33)(31,39)(32,34)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,e^2=h,f^2=i,b^a=be,d^a=di,e^a=eg,c^b=ci,e^b=eh,f^b=fgh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 70974);